Many 2-level polytopes from matroids

TL;DR

This paper explores the structural properties and enumeration of 2-level matroids, revealing a lower bound on the number of non-equivalent (n-1)-dimensional 2-level polytopes, thus advancing understanding of their combinatorial complexity.

Contribution

It introduces new structural insights into 2-level matroids and provides enumerative results, including a lower bound on the count of non-equivalent 2-level polytopes.

Findings

Number of non-equivalent (n-1)-dimensional 2-level polytopes grows exponentially with n.

Structural properties of 2-level matroids are characterized and utilized for enumeration.

Lower bound on the count of such polytopes is established.

Abstract

The family of 2-level matroids, that is, matroids whose base polytope is 2-level, has been recently studied and characterized by means of combinatorial properties. 2-level matroids generalize series-parallel graphs, which have been already successfully analyzed from the enumerative perspective. We bring to light some structural properties of 2-level matroids and exploit them for enumerative purposes. Moreover, the counting results are used to show that the number of combinatorially non-equivalent (n-1)-dimensional 2-level polytopes is bounded from below by $c \cdot n^{-5/2} \cdot \rho^{-n}$, where $c\approx 0.03791727 $ and $\rho^{-1} \approx 4.88052854$.